# Failure of Falconer distance problem in one dimension

I am told that the Falconer distance conjecture fails trivially in one dimension, but I really cannot find any reference for that. Precisely, I am asking the following question:

For a compact set $$E\subseteq \mathbb R^n$$, we define its distance set $$\Delta(E)\subseteq [0,\infty)$$ to be: $$\Delta(E)=\{|x-y|:x,y\in E\}.$$ Then if $$n=1$$, we ask the following questions?

1. Can we find a compact set $$E\subseteq [0,1]$$ such that $$\mathrm{dim}_H(E)>1/2$$ but $$\mathcal L^1(\Delta(E))=0$$?

2. Can we find a compact set $$E\subseteq [0,1]$$ such that $$\mathrm{dim}_H(E)=1$$ but $$\mathcal L^1(\Delta(E))=0$$?

• For an explicit example of a set satisfying 2, let $E$ be the set of points such that the $10^k$th digit of the decimal expansion is 0 for each $k$. – Anthony Quas Mar 7 at 16:38